Generation of Walsh functions
Contents
Program description
This applet allows to display the Hadamard matrices $\mathbf{H}_J$ for the construction of the Walsh functions $w_j$. The factor $J$ of the band spreading as well as the selection of the individual Walsh functions (by blue bordering the rows of the matrix) can be changed.
Theoretical background
Application
The Walsh functions are a group of periodic orthogonal functions. Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
- Due to their orthogonal properties and the favourable periodic cross-correlation function $\rm (PCCF)$, the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system. If you take any two lines and form the correlation (averaging over the products), the PCCF value is always zero.
- In asynchronous operation (example: uplink of a mobile radio system) or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading.
- In terms of $\rm (PACF)$ (periodic autocorrelation function) these sequences are less good: Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise $\rm (PN)$ sequence. That means: The synchronization is more difficult with Walsh functions than with PN sequences.
Construction
The construction of Walsh functions can be done recursively using the Hadamard matrices.
- A Hadamard matrix $\mathbf{H}_J$ of order $J$ is a $J\times J$ matrix, which contains line by line the $\pm 1$ weights of the Walsh sequences.
- The orders of the Hadamard matrices are fixed to powers of two, i.e. $J = 2^G$ applies to a natural number $G$. Starting from $\mathbf{H}_1 = [+1]$ and
- $$ \mathbf{H}_2 = \left[ \begin{array}{rr} +1 & +1\\ +1 & -1 \\ \end{array}\right] $$
the following relationship applies to the generation of further Hadamard matrices:
- $$ \mathbf{H}_{2N} = \left[ \begin{array}{rr} +\mathbf{H}_N & +\mathbf{H}_N\\ +\mathbf{H}_N & -\mathbf{H}_N \\ \end{array}\right] $$
$\text{Example:}$ The graphic shows the Hadamard matrix $\mathbf H_8$ (right) and the $J\hspace{-0.09cm} -\hspace{-0.09cm}1$ spreading sequences which can be constructed with it.
- Only $J\hspace{-0.09cm} -\hspace{-0.09cm}1$, because the unspreaded sequence $w_0(t)$ is usually not used.
- Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences $w_j(t)$.
- The submatrix $\mathbf H_4$ is highlighted in yellow.
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How to use the applet
(A) Selection of $G$ ⇒ Band spread factor: $J= 2^G$
(B) Selection of the Walsh function $w_j$ to be marked
About the authors
This interactive calculation tool was designed and realized at the Lehrstuhl für Nachrichtentechnik $\rm (LNT)$ of the Technical University of Munich $\rm (TUM)$.
- The first German version was created in 2007 by Thomas Großer in the context of his diploma thesis with "FlashMX–Actionscript" (Supervisor: Günter Söder).
- 2018/2019 the applet was converted on "HTML5" and redesigned by Carolin Mirschina (Engineering practice, supervisor: Tasnád Kernetzky ).
- 2020 this English version was made by Carolin Mirschina (working student) and Günter Söder. Translation using "www.DeepL.com/Translator" (free version).
